Extension without Cut

Lutz Straßburger

To cite this version:

Lutz Straßburger. Extension without Cut. Ann. Pure Appl. Logic, Elsevier, 2012, 163 (12),pp.1995-2007. <10.1016/j.apal.2012.07.004>. <hal-00759215>

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Extension without Cut

Lutz Straßburger

INRIA Saclay–Ile-de-France and Ecole Polytechnique, LIX, Rue de Saclay, 91128 Palaiseau Cedex, France

Abstract

In proof theory one distinguishes sequent proofs with cut and cut-free sequent proofs, while for proofcomplexity one distinguishes Frege-systems and extended Frege-systems. In this paper we show how deepinference can provide a uniform treatment for both classifications, such that we can define cut-free systemswith extension, which is neither possible with Frege-systems, nor with the sequent calculus. We show thatthe propositional pigeon-hole principle admits polynomial-size proofs in a cut-free system with extension. Wealso define cut-free systems with substitution and show that the cut-free system with extension p-simulatesthe cut-free system with substitution.

1. Introduction

For studying proof complexity (for propositional classical logic) one essentially distinguishes between twokinds of proof systems: Frege systems and extended Frege systems [1]. Roughly speaking, a Frege-systemconsists of a set of axioms and modus ponens, and in an extended Frege-system one can also use “abbrevi-ations”, i.e., fresh propositional variables abbreviating arbitrary formulas appearing in the proof. Clearly,any extended Frege-proof can be converted into a Frege-proof by systematically replacing the abbreviationsby the formulas they abbreviate, at the cost of an exponential increase of the size of the proof. Surprisingly,this distinction is not investigated from the proof theoretic viewpoint.

On the other hand, in proof theory one also distinguishes between two kinds of proof systems: those withcut and those without cut. In a well-designed proof system, it is always possible to convert a proof with cutsinto a cut-free proof, at the cost of an exponential increase of the size of the proof (see, e.g., [2]). The cutsare usually understood as “the use of auxiliary lemmas inside the proof”. The main tool for investigatingthe cut and its elimination from a proof is Gentzen’s sequent calculus [3].

The two proof classifications are usually not studied together. In fact, every Frege-system contains cutbecause of the presence of modus ponens. Hence, there is no such thing as a “cut-free Frege system”, or a“cut-free extended Frege-system”. Similarly, there are no “extended Gentzen systems”, because it does notmake sense to speak of abbreviations in the sequent calculus, where formulas are decomposed along theirmain connectives during proof search.1 This can be summarized by the classification of proof systems shownin Figure 1, where S1 ⊆ S2 means that S2 includes S1 (and therefore S2 p-simulates S1).

cut-freesequent systems

⊆sequent systemswith cut

=Fregesystems

⊆extendedFrege systems

Figure 1: Classification of proof systems

There are classes of tautologies that admit no polynomial size proofs in cut-free sequent calculus [5](and related systems, like resolution [6] and tableaux). But no such class is known for systems with cut

1The extension discussed in this paper should not be confused with the notion of “definition” in the sequent calculusLKDe [4], in which the abbreviation may occur in the endsequent of the proof.

Preprint submitted to Elsevier July 31, 2012

systems with cut (without extension) ⊆ systems with cut and extension

∪pp ∪pp

cut-free systems (without extension) ⊆ cut-free systems with extension

Figure 2: Refined classification of proof systems

or for extended Frege systems. The question whether there is a short, i.e., polynomial size, proof of everytautology A is equivalent to the question whether NP is equal to coNP.

The contributions of this paper can be summarized as follows:

(i) I provide a deductive system in which extension is independent from the cut, i.e., we can now studycut-free systems with extension.2 Figure 2 shows the refined classification of proof systems. I use asformalism the calculus of structures [9, 10, 11]. Thus, this paper is a continuation of the work byBruscoli and Guglielmi [12], who observed that by using deep inference one can bring the extensionrule to a deductive formalism which has originally been designed to study cut-elimination.

(ii) I also present a cut-free system with substitution and investigate the relation between substitution andextension in a cut-free setting.

(iii) In order to provide evidence that it indeed makes sense to study extension (or substitution) indepen-dently from cut, I will show polynomial-size proofs for the propositional pigeon-hole principle (PHP)without cut. At the same time, I propose a new class of tautologies (called QHQ), that have similarproperties as the PHP wrt. proof complexity, but have the additional property of being balanced, i.e.,every atom occurs exactly once positive and exactly once negative.

Sections 2–4 of this paper contain preliminaries on proof systems in general and the calculus of structuresin particular. Then, Sections 5–7 are dedicated to points (i)–(iii) above.

2. Preliminaries on Proof Systems

Following [1], we define a proof system to be a surjective PTIME-function S : Σ∗ → T where Σ is somefinite alphabet (and Σ∗ the set of all finite words over Σ) and T is the set of all Boolean tautologies. Anelement of π ∈ Σ∗ is called a proof and S(π) its conclusion. We denote by |π| the size of π, i.e., the numberof symbols in π. Given two proof systems S1 : Σ

∗1 → T and S2 : Σ

∗2 → T , we say that S2 p-simulates S1 iff

there is a polynomial p such that for every proof π1 ∈ Σ∗1 there is a proof π2 ∈ Σ∗

2 of the same conclusion(i.e., S2(π2) = S1(π1)) such that |π2| ≤ |p(π1)|. We say that S1 and S2 are p-equivalent iff they p-simulateeach other. A proof system S : Σ∗ → T is polynomially bounded iff there is a polynomial p such that forevery tautology T ∈ T there is a proof π ∈ Σ∗ with S(π) = T and |π| ≤ p(|T |). If |π| ≤ p(|T |) for someπ ∈ Σ∗ with S(π) = T , we also say that π is a short proof of T . Thus, a polynomially bounded proof systemis one in which every tautology has a short proof wrt. some fixed bounding polynomial p. The questionwhether there is a polynomially bounded proof system is equivalent to the question whether the complexityclass NP is closed under complement:

Theorem 2.1. There exists a polynomially bounded proof system if and only if NP = coNP. [1]

2Technically speaking, Haken’s extended resolution [6] is a cut-free system with extension, but his system is not suited tostudy cut elimination. And it is not clear how to incorporate Haken’s extension into the recent work on resolution and cutelimination [7, 8].

2

For the sake of simplicity, let us consider for the rest of the paper only formulas in negation normal form.More precisely, formulas are generated from a countable set A = {a, b, c, . . .} of propositional variablesand their negations A = {a, b, c, . . .} via the binary connectives ∧ and ∨, called and and or, respectively.3

I denote formulas by capital Latin letters (A,B,C, . . .). Negation is defined for all formulas via the de

Morgan laws: ¯a = a and A ∧ B = B ∨ A and A ∨ B = B ∧ A. It follows immediately that ¯A = A for allformulas A. The elements of the set A ∪ A are also called literals. We can write A⇒B for A ∨ B and A⇔Bfor [A ∨ B] ∧ [B ∨ A].

3. Preliminaries on the Calculus of Structures

I assume the reader to be familiar with sequent calculus or natural deduction systems [3], in whichinference rules decompose formulas along their main connectives. On the other hand, in the calculus ofstructures [9, 10, 11], inference rules are allowed to do arbitrary rewriting deep inside formulas. In thispaper, I use the following rule schemes (to be applied on formulas in negation normal form):

F{B}ai↓ −−−−−−−−−−−−−−−−

F{B ∧ [a ∨ a]}

F{A ∧ [B ∨ C]}s −−−−−−−−−−−−−−−−−−F{(A ∧ B) ∨ C}

F{B}w↓ −−−−−−−−−−−

F{B ∨ A}

F{a ∨ a}ac↓ −−−−−−−−−−

F{a}

F{(A ∧ B) ∨ (C ∧ D)}m −−−−−−−−−−−−−−−−−−−−−−−−−−

F{[A ∨ C] ∧ [B ∨ D]}

(1)

where A, B, C, and D must be seen as formula variables, and a is a propositional variable or its negation.These rules are called (atomic) identity, switch, weakening, (atomic) contraction, and medial, respectively.The rules in (1) are written in the style of inference rule schemes in proof theory but they behave as rewriterules in term rewriting, i.e., they can be applied deep inside any (positive) formula context F{ }. To easereadability of large formulas, I will use [ ] for brackets around disjunctions and ( ) for brackets aroundconjunctions. The rewriting rules in (1) are applied modulo associativity and commutativity for ∧ and ∨.More precisely, we will do rewriting modulo the equational theory generated by

A ∧ (B ∧ C) = (A ∧ B) ∧ C A ∧ B = B ∧ A

A ∨ [B ∨ C] = [A ∨ B] ∨ C A ∨ B = B ∨ A(2)

Because of this, we can systematically omit superfluous parentheses in order to ease readability; e.g., insteadof A ∧ ((B ∧ C) ∧ D) we write A ∧ B ∧ C ∧ D. A derivation is a rewrite path via (1) modulo (2). Here is anexample:

([b ∨ b] ∧ [b ∨ b]) ∨ (b ∧ [a ∨ a])2∗ac↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

([b ∨ b] ∧ b) ∨ (b ∧ a)ai↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

([b ∨ b] ∧ (b ∧ [a ∨ a])) ∨ (b ∧ a)s −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−([b ∨ b] ∧ [(b ∧ a) ∨ a]) ∨ (b ∧ a)

= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−([b ∨ b] ∧ [a ∨ (a ∧ b)]) ∨ (b ∧ a)

ac↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−(b ∧ [a ∨ (a ∧ b)]) ∨ (b ∧ a)

s −−−−−−−−−−−−−−−−−−−−−−−−−−−−−[(b ∧ a) ∨ (a ∧ b)] ∨ (b ∧ a)

(3)

The notation n∗r is used to indicate that there are n applications of the rule r. In the hope of helping thereader, I sometimes use a “fake inference rule”

A= −−

B(4)

governed by the side condition that A = B under the equivalence relation generated by (2).

3For simplicity, I do not introduce special symbols for the units truth and falsum. Note that these units can be recoveredby the formulas p0 ∨ p0 and p0 ∧ p0, respectively, where p0 is a fresh propositional variable.

3

Remark 3.1. Instead of doing rewriting modulo, one could equivalently add four inference rules

F{[A ∨ B] ∨ C}=1 −−−−−−−−−−−−−−−−−−

F{A ∨ [B ∨ C]}

F{(A ∧ B) ∧ C}=2 −−−−−−−−−−−−−−−−−−

F{A ∧ (B ∧ C)}

F{A ∨ B}=3 −−−−−−−−−−−

F{B ∨ A}

F{A ∧ B}=4 −−−−−−−−−−−

F{B ∧ A}(5)

Computationally there is no difference between the two approaches since the the equivalence modulo = canbe checked in time O(n log n).

In order to obtain proofs without hypotheses, we need an axiom, which is in our case just a variant ofthe rule ai↓:

ai↓ −−−−−a ∨ a

(6)

The rule in (6) cannot be applied inside a context F{ }, but it is in spirit the same rule as ai↓ in (1), and Iuse therefore the same name. Given a system S, we write

A

S

∥

∥

∥

∥

∥

∥π1

B

and

−S

∥

∥

∥

∥

∥

∥π2

B

to denote a derivation π1 in the system S from premise A to conclusion B, and a proof π2 in the system S

without premise and with conclusion B, respectively. I write KS to denote the system shown in (1), togetherwith the rule in (6). A proof in KS uses the axiom (6) exactly once.

Remark 3.2. The original formulation of KS in [11] uses explicit units t and f for truth and falsum,respectively, and thus contains more rules and equations to deal with them. If we denote by KS+ the systemwith units, and by KS− the system without units, then we have that KS+ and KS− are p-equivalent underthe translation mentioned in Footnote 3. This can easily be shown by using the equations B ∧ t = B = B ∨ f

and B ∧ f = f and B ∨ t = t (see also [13]). Since many Frege systems are given without explicit units, andGentzen’s original LK comes without units, it might be helpful to see a presentation of KS that does notrely on the presence of units. Anyhow, everything that is said in this paper does also hold for the variant ofKS with units.

The following two propositions have first been proved in [11]:

Proposition 3.3. The rules

i↓ −−−−−−A ∨ A

F{B}i↓ −−−−−−−−−−−−−−−−−−

F{

B ∧ [A ∨ A]}

F{A ∨ A}c↓ −−−−−−−−−−−

F{A}

F{[A ∨ C] ∧ [B ∨ C]}d↓ −−−−−−−−−−−−−−−−−−−−−−−−

F{(A ∧ B) ∨ C}

are derivable in KS. More precisely, KS p-simulates KS ∪ {i↓, c↓, d↓}.

The rules i↓ and c↓ are the general (non-atomic) versions of ai↓ and ac↓, respectively.

Proposition 3.4. The system KS p-simulates cut-free sequent calculus.

The converse is not true, i.e., cut-free sequent calculus cannot p-simulate KS. A counter-example canbe found in [12], where Bruscoli and Guglielmi show that the example used by Statman [5] to prove anexponential lower bound for cut-free sequent calculus admits polynomial size proofs in KS. This situationchanges when we add the cut rule, which is dual to the identity rule

F{(a ∧ a) ∨ B}ai↑ −−−−−−−−−−−−−−−−−

F{B}. (7)

The system KS∪{ai↑} will in the following be denoted by SKS. The following two propositions are also dueto [11]:

4

Proposition 3.5. The rules

F{

(A ∧ A) ∨ B}

i↑ −−−−−−−−−−−−−−−−−−−F{B}

F{A}c↑ −−−−−−−−−−−

F{A ∧ A}

F{A ∧ B}w↑ −−−−−−−−−−−

F{B}(8)

are derivable in SKS. More precisely, SKS p-simulates SKS ∪ {i↑, c↑,w↑}.

Proposition 3.6. SKS is p-equivalent to every sequent system with cut.

Finally, let us mention the following theorem, stating soundness, completeness, cut elimination, and thededuction theorem for KS.

Theorem 3.7. For any formulas A and B, we have:

The formula A⇒B isa valid implication.

iff

−KS

∥

∥

∥

∥

∥

∥

A ∨ Biff

A

SKS

∥

∥

∥

∥

∥

∥

B

This does not only hold for classical logic, but also for linear logic and modal logic (for a proof see [14, 15]).

4. Relation between Calculus of Structures and Frege Systems

Frege systems (also known as Hilbert systems or Hilbert-Frege-systems or Hilbert-Ackermann-systems [16,17]), consist of a set of axioms (more precisely, axiom schemes) and a set of inference rules, which in thecase of classical propositional logic only contains modus ponens:

A A ⇒ Bmodus ponens −−−−−−−−−−−−−−−−−

B

I assume the reader to be familiar with Frege systems, and I will not go into further details. The importantfacts are that the set of axioms in a Frege system has to be sound and complete, and that all Frege systemsp-simulate each other [1]. We also immediately have the following theorem:

Theorem 4.1. SKS is p-equivalent to every Frege-system.

This follows immediately from Proposition 3.6 and a result by [1]. In [12] one can find a direct proof.Because it will be needed later, I sketch here the basic idea. For p-simulating a Frege system F with SKS,we first exhibit an SKS proof for every axiom in F. Then we proceed by induction on the length of the proofπ in F and keep all formulas appearing in π in a conjunction F1 ∧ F2 ∧ · · · ∧ Fn. Now we can simulate modusponens:

A A ∨ Bmodus ponens −−−−−−−−−−−−−−

B;

A ∧ [A ∨ B]s −−−−−−−−−−−−−(A ∧ A) ∨ B

i↑ −−−−−−−−−−−−−B

Note that we might need to duplicate a formula Fi by using c↑. Finally we remove the superfluous copiesby using w↑. Conversely, we show that a Frege system can p-simulate SKS by exhibiting for every rule

Ar −−B

a Frege-proof of A ∨ B. Then we show by induction that for every context F{ } also F{A} ∨ F{B} has aFrege proof. Then the application of an inference rule in SKS can be simulated by modus ponens.

5

5. Extension

Let us now turn to the actual interest of this paper, the extension rule (first formulated by Tseitin [18]),which allows us to use abbreviations in the proof. I.e., there is a finite set of fresh and mutually distinctpropositional variables a1, . . . , an which can abbreviate formulas A1, . . . , An, that obey the side conditionthat for all 1 ≤ i ≤ n, the variable ai does not appear in A1, . . . , Ai. Extension can easily be integratedin a Frege-system by simply adding the formulas ai ⇔ Ai, for 1 ≤ i ≤ n, to the set of axioms. In thatcase we speak of an extended Frege-system [1]. In the sequent calculus one could add these formulas asnon-logical axioms, with the consequence that cut-elimination would not hold anymore. This very idea isused by Bruscoli and Guglielmi in [12] for adding extension to a system in the calculus of structures: insteadof starting a proof from no premises, they use the conjunction

[a1 ∨ A1] ∧ [A1 ∨ a1] ∧ · · · ∧ [an ∨ An] ∧ [An ∨ an] (9)

of all extension formulas as premise. Let us write xSKS to denote the system SKS with the extensionincorporated this way, i.e., a proof of a formula B in xSKS is a derivation

[a1 ∨ A1] ∧ [A1 ∨ a1] ∧ · · · ∧ [an ∨ An] ∧ [An ∨ an]

SKS

∥

∥

∥

∥

∥

∥π

B

(10)

wherethe propositional variables a1, . . . , an are mutually distinct, and forall 1 ≤ i ≤ n, the variable ai does not appear in A1, . . . , Ai nor in B.

(11)

Theorem 5.1. xSKS is p-equivalent to every extended Frege-system.

The proof can be found in [12], and is almost literally the same as for Theorem 4.1.It should be clear that xSKS crucially relies on the presence of cut, in the same way as extended Frege-

system rely on the presence of modus ponens: The premise of (10) contains the variables a1, . . . , an, whichdo not appear in the conclusion B. Thus, the derivation in (10) must contain cuts. This raises the questionwhether the virtues of extension can also be used in a cut-free system.4 For this, let us for every extensionaxiom ai ⇔ Ai add the following two rules (we use the same name for both of them):

F{ai}ext↓ −−−−−−−

F{Ai}and

F{ai}ext↓ −−−−−−−−

F{

Ai

} (12)

We write eKS to denote the system KS ∪ {ext↓} and we write eSKS for SKS ∪ {ext↓}. Note that the ruleext↓ is not sound. Consider for example the extension axiom a⇔ b ∧ c where a abbreviates b ∧ c. Applying itto a ∨ a (which is a tautology) yields (b ∧ c) ∨ a (which is not a tautology). Nonetheless, we allow to apply(12) in an arbitrary context F{ }, provided that condition (11) is satisfied. Then we have the following:

Theorem 5.2. The systems eKS and eSKS are sound and complete for classical propositional logic.

Proof. Completeness of both systems follows from completeness of KS, and soundness of eSKS follows fromTheorem 5.3 below and Theorem 5.1 above. This entails soundness of eKS.

Theorem 5.3. The systems eSKS and xSKS are p-equivalent.

4One could allow to add a disjunction of formulas ai ∧ ai to the conclusion, in the same way as we add a conjunction of[ai ∨ Ai] ∧ [Ai ∨ ai] to the premise [12]. Some readers might consider this to be cut-free, but the question remains whether wecan obtain cut-freeness without changing the notions of derivation and proof.

6

Proof. Given a proof π of a formula B in xSKS, we can construct

−{ai↓}

∥

∥

∥

∥

∥

∥π2

[a1 ∨ a1] ∧ [a1 ∨ a1] ∧ · · · ∧ [an ∨ an] ∧ [an ∨ an]

{ext↓}

∥

∥

∥

∥

∥

∥π1

[a1 ∨ A1] ∧ [A1 ∨ a1] ∧ · · · ∧ [an ∨ An] ∧ [An ∨ an]

SKS

∥

∥

∥

∥

∥

∥π

B

(13)

where π1 consists of 2n instances of ext↓ and π2 of 2n instances of ai↓. Hence, eSKS p-simulates xSKS. Forthe converse, assume we have an eSKS proof π of a formula B. We can put every line of π in conjunctionwith the formula (9), and add a coweakening w↑ (see Proposition 3.5) at the bottom:

−eSKS

∥

∥

∥

∥

∥

∥π

B;

[a1 ∨ A1] ∧ [A1 ∨ a1] ∧ · · · ∧ [an ∨ An] ∧ [An ∨ an]

eSKS

∥

∥

∥

∥

∥

∥π′

[a1 ∨ A1] ∧ [A1 ∨ a1] ∧ · · · ∧ [an ∨ An] ∧ [An ∨ an] ∧ Bw↑ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

B

The instances of ext↓ in π′ can now be removed as follows:

· · · ∧ [ai ∨ Ai] ∧ · · · ∧ F{ai}ext↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

· · · ∧ [ai ∨ Ai] ∧ · · · ∧ F{Ai};

· · · ∧ [ai ∨ Ai] ∧ · · · ∧ F{ai}c↑ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

· · · ∧ [ai ∨ Ai] ∧ [ai ∨ Ai] ∧ · · · ∧ F{ai}

{s}

∥

∥

∥

∥

∥

∥πs

· · · ∧ [ai ∨ Ai] ∧ · · · ∧ F{ai ∧ [ai ∨ Ai]}s −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−· · · ∧ [ai ∨ Ai] ∧ · · · ∧ F{(ai ∧ ai) ∨ Ai}

ai↑ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−· · · ∧ [ai ∨ Ai] ∧ · · · ∧ F{Ai}

(14)

where F{ } is an arbitrary (positive) context, and the existence of πs (which contains only instances of therule s) can be shown by an easy induction on F{ } (see e.g, Lemma 4.3.20 in [15]). The length of πs isbound by the depth of F{ }. Note the crucial use of the cut rule in (14).

System eKS gives us a way of adding extension to a deductive system independently from cut. To showthat extension without cut is potentially useful for giving short proofs for some of the standard benchmarktautologies, we give in Section 7 polynomial size proofs of the propositional pigeon hole principle in eKS.

Remark 5.4. When we say “independent from cut”, we have to clarify what “cut” means. The way weadded extension to KS to get eKS is clearly independent from the chosen language. E.g., it does not matterwhether we chose a presentation with or without units: if we define eKS− and eKS+ by adding ext↓ to KS−

and KS+, respectively (see Remark 3.2), then one can easily show that eKS− and eKS+ p-simulate each other(if the units appear inside the extension formulas Ai then they can be removed using B ∧ t = B = B ∨ f

and B ∧ f = f and B ∨ t = t, and if Ai = t or Ai = f then this extension axiom can be eliminated withoutincreasing the size of the proof). However, the power of the rule ext↓ depends on the other rules that arepresent. For example, in order to make Theorem 5.3 work, the rules ai↓ and s should be derivable in thechosen system. Furthermore, for the polynomial size proofs of the propositional pigeon hole principle inSection 7 we will need associativity and commutativity of conjunction. More precisely, for completeness ofKS only the rules =1 and =3 in (5) would be necessary [19], but for Section 7 we also need rules =2 and =4

(see Remark 3.1).

6. Substitution

Let us next consider systems with substitution. A substitution is a function σ from the set A ofpropositional variables to the set F of formulas, such that σ(a) = a for almost all a ∈ A . We can define

7

σ(A) inductively for all formulas via σ(A ∧ B) = σ(A) ∧ σ(B) and σ(A ∨ B) = σ(A) ∨ σ(B) and σ(A) = σ(A).Following the tradition, we write Aσ for σ(A). For example, if A = a ∨ b ∨ b and σ = {a 7→ a ∧ b, b 7→ a ∨ c}then Aσ = (a ∧ b) ∨ (a ∧ c) ∨ a ∨ c. We can define the inference rule for substitution

Asub↓ −−−

Aσ(15)

Note that the rule sub↓ cannot be applied inside a context F{ }. Thus, it is exactly the same rule as inFrege systems and in strong contrast to all other rules in deep inference. Let us define sSKS = SKS∪{sub↓}and sKS = KS ∪ {sub↓}. The following has been proved in [12]:

Theorem 6.1. sSKS is p-equivalent to any Frege-system with substitution.

However, contrary to the previous cases, there is no immediate easy proof of Theorem 6.1, because thesubstitution rule is stronger in Frege systems than in SKS. The reason is that in Frege systems one can, aftera substitution σ has been applied to a formula A, reuse the original A as well as the substituted version Aσ.

Thus, to prove Theorem 6.1, Bruscoli and Guglielmi use in [12] the result by Krajıcek and Pudlak [20]on the p-equivalence of Frege-system with extension and Frege-system with substitution.

We give here a direct proof of the p-equivalence of sSKS and xSKS (and eSKS).

Theorem 6.2. sSKS p-simulates xSKS.

Proof. This proof can already be found in [12]. For a given xSKS proof π of a formula B, we construct

i↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−(an ∧ An) ∨ (An ∧ an) ∨ · · · ∨ (a1 ∧ A1) ∨ (A1 ∧ a1) ∨ ([a1 ∨ A1] ∧ [A1 ∨ a1] ∧ · · · ∧ [an ∨ An] ∧ [An ∨ an])

SKS

∥

∥

∥

∥

∥

∥π′

(an ∧ An) ∨ (An ∧ an) ∨ · · · ∨ (a1 ∧ A1) ∨ (A1 ∧ a1) ∨ B= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

(an ∧ An) ∨ (An ∧ an) ∨ (an−1 ∧ An−1) ∨ (An−1 ∧ an−1) ∨ · · · ∨ (a1 ∧ A1) ∨ (A1 ∧ a1) ∨ Bsub↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

(An ∧ An) ∨ (An ∧ An) ∨ (an−1 ∧ An−1) ∨ (An−1 ∧ an−1) ∨ · · · ∨ (a1 ∧ A1) ∨ (A1 ∧ a1) ∨ B2∗i↑ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

(an−1 ∧ An−1) ∨ (An−1 ∧ an−1) ∨ · · · ∨ (a1 ∧ A1) ∨ (A1 ∧ a1) ∨ Bsub↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

...2∗i↑ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−

(a1 ∧ A1) ∨ (A1 ∧ a1) ∨ Bsub↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

(A1 ∧ A1) ∨ (A1 ∧ A1) ∨ B2∗i↑ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

B

(16)

where π′ is obtained from π by putting every formula in disjunction with

(an ∧ An) ∨ (An ∧ an) ∨ · · · ∨ (a1 ∧ A1) ∨ (A1 ∧ a1)

The derivation (16) is a valid derivation in sSKS because of condition (11). Note that we proceed backwardsin eliminating the ai in order to keep the size of the proof polynomial.

For the other direction, the basic idea is to simulate the substitution inference step from A to Aσ bymany extension inference steps, one for each occurrence of a variable a with σ(a) 6= a in A. Consider forexample:

−∥∥

∥

∥

∥

∥π2

F{a ∨ (b ∧ c) ∨ a}sub↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

F{(a ∧ c) ∨ (b ∧ [a ∨ c]) ∨ a ∨ c}∥

∥

∥

∥

∥

∥π1

B

;

−∥∥

∥

∥

∥

∥π2

F{a ∨ (b ∧ c) ∨ a}ext↓ −−−−−−−−−−−−−−−−−−−−−−−−−−

F{(a ∧ c) ∨ (b ∧ c) ∨ a}ext↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

F{(a ∧ c) ∨ (b ∧ [a ∨ c]) ∨ a}ext↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

F{(a ∧ c) ∨ (b ∧ [a ∨ c]) ∨ a ∨ c}∥

∥

∥

∥

∥

∥π1

B

(17)

8

where the used substitution is {a 7→ a ∧ c, c 7→ a ∨ c} and the context F{ } does not contain any occurrencesof a or c. The problem with this is that the result will, in general, not be a valid proof because bothconditions in (11) might be violated. For this reason we first have to rename the variables a and c in π2:

−∥∥

∥

∥

∥

∥π′

2

F{a′ ∨ (b ∧ c′) ∨ a′}sub↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

F{(a ∧ c) ∨ (b ∧ [a ∨ c]) ∨ a ∨ c}∥

∥

∥

∥

∥

∥π1

B

;

−∥∥

∥

∥

∥

∥π′

2

F{a′ ∨ (b ∧ c′) ∨ a′}ext↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−

F{(a ∧ c) ∨ (b ∧ c′) ∨ a′}ext↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

F{(a ∧ c) ∨ (b ∧ [a ∨ c]) ∨ a′}ext↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

F{(a ∧ c) ∨ (b ∧ [a ∨ c]) ∨ a ∨ c}∥

∥

∥

∥

∥

∥π1

B

(18)

Here a and c have been replaced everywhere in π2 by fresh variables a′ and c′, respectively. The new sub-stitution is {a′ 7→ a ∧ c, c′ 7→ a ∨ c}, which can be replaced by instances of extension, without violating (11).

Theorem 6.3. eSKS p-simulates sSKS.

Proof. Let π be an sSKS proof of a formula B. Suppose π contains k instances of sub↓, and let σ1,1, . . . , σk,1

be the k substitutions used in them. Then π is of the shape as shown in the left-most derivation in Figure 3.In the following, we use Ai,j to denote the set of variables a with σi,j(a) 6= a. As explained above, we cannow iteratively rename the propositional variables in A1,1, . . . ,Ak,1, starting from the bottommost instanceof sub↓, as indicated in Figure 3. The result of this renaming is shown in the rightmost derivation in Figure 3and has the property that

for all i with 1 ≤ i ≤ k, we have that no variable in Ai,i+1

appears in any of π1,1, π2,2, . . . , πi,i.(19)

Let Ai,i+1 = {ai,1, . . . , ai,mi}, and let Ai,h = σi,i+1(ai,h). We now have n = m1 +m2 + · · ·+mk extension

variables, defined viaai,h

ext↓ −−−−−−Ai,h

andai,h

ext↓ −−−−−−Ai,h

If we give the index pair (i, h) the lexicographic ordering, it immediately follows from (19) that condition (11)is fulfilled. Hence, we can trivially replace each instance of sub↓ by a sequence of instances of ext↓, whosenumber is bound by the size of the Bi,i+1. Hence, the size of the resulting eSKS proof is at most quadraticin the size of π.

Note that the transformation in the proof of Theorem 6.3 does not involve any cuts. Hence, we havealso proved the following:

Theorem 6.4. eKS p-simulates sKS.

Remark 6.5. The possible interest of sKS has already been mentioned in [12], but up to now it is unknownwhether sKS can p-simulate eKS, although I conjecture this to be the case.

7. Pigeonhole Principle and Balanced Tautologies

In this section we see two classes of tautologies which both admit polynomial-size proofs in eKS and sKS.The first one is the propositional pigeon-hole principle. The second one is a variation which has the propertythat every member of the class is a balanced tautology. A formula A is balanced if every propositional variableoccurring in A occurs exactly twice, once positive and once negated. For example,

([a ∨ b] ∧ [d ∨ e]) ∨(

[a ∨ c] ∧[

d ∨ f])

∨([

b ∨ c]

∧[

e ∨ f])

9

−SKS

∥

∥

∥

∥

∥

∥πk+1,1

Bk,1sub↓ −−−−−−−−−

Bk,1σk,1

SKS

∥

∥

∥

∥

∥

∥πk,1

...

SKS

∥

∥

∥

∥

∥

∥π3,1

B2,1sub↓ −−−−−−−−−

B2,1σ2,1

SKS

∥

∥

∥

∥

∥

∥π2,1

B1,1sub↓ −−−−−−−−−

B1,1σ1,1

SKS

∥

∥

∥

∥

∥

∥π1,1

B

;

−SKS

∥

∥

∥

∥

∥

∥πk+1,2

Bk,2sub↓ −−−−−−−−−

Bk,2σk,2

SKS

∥

∥

∥

∥

∥

∥πk,2

...

SKS

∥

∥

∥

∥

∥

∥π3,2

B2,2sub↓ −−−−−−−−−

B2,2σ2,2

SKS

∥

∥

∥

∥

∥

∥π2,2

B1,2sub↓ −−−−−−−−−

B1,2σ1,2= −−−−−−−−−

B1,1σ1,1

SKS

∥

∥

∥

∥

∥

∥π1,1

B

;

−SKS

∥

∥

∥

∥

∥

∥πk+1,3

Bk,3sub↓ −−−−−−−−−

Bk,3σk,3

SKS

∥

∥

∥

∥

∥

∥πk,3

...

SKS

∥

∥

∥

∥

∥

∥π3,3

B2,3sub↓ −−−−−−−−−

B2,3σ2,3= −−−−−−−−−

B2,2σ2,2

SKS

∥

∥

∥

∥

∥

∥π2,2

B1,2sub↓ −−−−−−−−−

B1,2σ1,2= −−−−−−−−−

B1,1σ1,1

SKS

∥

∥

∥

∥

∥

∥π1,1

B

; · · · ;

−SKS

∥

∥

∥

∥

∥

∥πk+1,k+1

Bk,k+1sub↓ −−−−−−−−−−−−−−−

Bk,k+1σk,k+1= −−−−−−−−−−−−−−−

Bk,kσk,k

SKS

∥

∥

∥

∥

∥

∥πk,k

...

SKS

∥

∥

∥

∥

∥

∥π3,3

B2,3sub↓ −−−−−−−−−

B2,3σ2,3= −−−−−−−−−

B2,2σ2,2

SKS

∥

∥

∥

∥

∥

∥π2,2

B1,2sub↓ −−−−−−−−−

B1,2σ1,2= −−−−−−−−−

B1,1σ1,1

SKS

∥

∥

∥

∥

∥

∥π1,1

B

Figure 3: Renaming propositional variables in an sSKS proof

is balanced (and a tautology), whereas

a ∨ a ∨ (a ∧ a) and a ∧ a ∧ b

are not balanced. I use the notation∧∧

0≤i≤n Fi as abbreviation for F0 ∧ · · · ∧ Fn, and similarly for∨∨

.Furthermore, for a literal a, I abbreviate the formula a ∨ · · · ∨ a by an, if there are n copies of a. Considernow

PHPn =∧∧

0≤i≤n

∨∨

1≤j≤n

pi,j ⇒∨∨

0≤i<m≤n

∨∨

1≤j≤n

(pi,j ∧ pm,j) (20)

This formula is called the propositional pigeon hole principle because it expresses the fact that if there aren + 1 pigeons and only n holes and every pigeon is in a hole then at least one hole contains two pigeons,provided one reads the propositional variable pi,j as “pigeon i sits in hole j”.

The formulas (20) have been well investigated from the viewpoint of proof complexity. In [1] they wherepresented as a candidate for separating Frege systems and extended Frege systems (wrt. p-simulation). ButBuss [21] has shown that PHPn admits a polynomial-size proof in a Frege system (and therefore in SKS)for every n.

I will here show that in eKS as well as in sKS we have cut-free polynomial-size proofs for (20). For thisI use a new class of tautologies which also admit polynomial-size proofs in eKS, and which are defined asfollows:

QHQn =∨∨

0≤i≤n

∧∧

1≤j≤n

[

∨∨

1≤k≤i

qi,j,k ∨∨∨

i<k≤n

qk,j,i+1

]

(21)

10

Here are the first three examples:

QHQ1 = q1,1,1 ∨ q1,1,1

QHQ2 = ([q1,1,1 ∨ q2,1,1] ∧ [q1,2,1 ∨ q2,2,1]) ∨ ([q1,1,1 ∨ q2,1,2] ∧ [q1,2,1 ∨ q2,2,2])∨

([q2,1,1 ∨ q2,1,2] ∧ [q2,2,1 ∨ q2,2,2])

QHQ3 = ([q1,1,1 ∨ q2,1,1 ∨ q3,1,1] ∧ [q1,2,1 ∨ q2,2,1 ∨ q3,2,1] ∧ [q1,3,1 ∨ q2,3,1 ∨ q3,3,1])∨

([q1,1,1 ∨ q2,1,2 ∨ q3,1,2] ∧ [q1,2,1 ∨ q2,2,2 ∨ q3,2,2] ∧ [q1,3,1 ∨ q2,3,2 ∨ q3,3,2])∨

([q2,1,1 ∨ q2,1,2 ∨ q3,1,3] ∧ [q2,2,1 ∨ q2,2,2 ∨ q3,2,3] ∧ [q2,3,1 ∨ q2,3,2 ∨ q3,3,3])∨

([q3,1,1 ∨ q3,1,2 ∨ q3,1,3] ∧ [q3,2,1 ∨ q3,2,2 ∨ q3,2,3] ∧ [q3,3,1 ∨ q3,3,2 ∨ q3,3,3])

The tautologies QHQn are balanced. This means that the size of a proof in KS (or related systems) ofsuch a tautology is directly related to the number of applications of ac↓. Furthermore, all proofs that weshow here do not contain any weakening. This makes this class interesting for investigating the gap betweenlinear logic and classical logic [22, 23].

The formulas QHQ1 and QHQ2 are easily provable in KS \ {ac↓}. One might be tempted to conjecturethat KS \ {ac↓} or eKS \ {ac↓} is already complete for the class of balanced tautologies. But unfortunately,this is not the case. The smallest counterexample known to me is QHQ3. Every possible application of ai↓,s, m, or w↓ leads to a non-tautologous formula. Thus also the extension rule is of no use. (The same is truefor all formulas QHQn with n ≥ 3.)

This is not surprising under the view of the following theorem, which says that balanced tautologies arenot easier to prove than other tautologies.

Theorem 7.1. The set of balanced tautologies is coNP-complete.

Proof. We can reduce provability of general tautologies to provability of balanced tautologies. For a for-mula B, we let B′ be the formula obtained from B by doing the following replacement for every propositionalvariable a occurring in B: Let n be the number of occurrences of a in positive form in B, and let m be thenumber of occurrences of a in B. If n ≥ 1 and m ≥ 1, then introduce n ·m fresh propositional variables ai,jfor 1 ≤ i ≤ n and 1 ≤ j ≤ m. Now replace for every 1 ≤ i ≤ n the ith occurrence of a by ai,1 ∨ · · · ∨ ai,m,and replace for every 1 ≤ j ≤ m the jth occurrence of a by a1,j ∨ · · · ∨ an,j . If n = 0, then introduce m freshvariables a1, . . . am and replace the jth a by aj ∧ aj . If m = 0, proceed similarly (cf. Footnote 3). Then B′

is balanced, and its size is quadratic in the size of B. Furthermore, B′ is a tautology if and only if B is atautology. This can be seen as follows: Any KS-proof of B can be transformed into a KS-proof of B′ bypropagating the replacements of literals through the proof; atomic contractions ac↓ are replaced by generalcontractions c↓, and identities ai↓ by weakenings and identities:

F{B}ai↓ −−−−−−−−−−−−−−−−

F{B ∧ [a ∨ a]};

F{B}ai↓ −−−−−−−−−−−−−−−−−−−−−−

F{B ∧ [ai,j ∨ ai,j ]}(m+n−2)∗w↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

F{B ∧ [a1,j ∨ · · · ∨ an,j ∨ ai,1 ∨ · · · ∨ ai,m]}

Conversely, an SKS-proof of B′ can be transformed into a SKS-proof of B by forgetting the indices andadding a sufficient number of contractions c↓ and coweakenings w↑

F{a ∨ · · · ∨ a}

c↓

∥

∥

∥

∥

∥

∥

F{a}

F{a ∨ · · · ∨ a}

c↓

∥

∥

∥

∥

∥

∥

F{a}

F{a ∧ a}w↑ −−−−−−−−−−

F{a}

F{a ∧ a}w↑ −−−−−−−−−−

F{a}

at the bottom of the derivation. (Thus, by coweakening-elimination, also a KS-proof of B′ can be transformedinto a KS-proof of B.)

Let us now reduce PHPn to QHQn. We first replace the implication by disjunction and negation, and

11

then apply associativity and commutativity of ∨:

PHPn =∨∨

0≤i≤n

∧∧

1≤j≤n

pi,j ∨∨∨

0≤i<n

∨∨

i<m≤n

∨∨

1≤j≤n

(pi,j ∧ pm,j)

=∨∨

0≤i≤n

∧∧

1≤j≤n

pi,j ∨∨∨

0≤i≤n

∨∨

1≤j≤n

∨∨

i<m≤n

(pi,j ∧ pm,j)

=∨∨

0≤i≤n

[

∧∧

1≤j≤n

pi,j ∨∨∨

1≤j≤n

∨∨

i<m≤n

(pi,j ∧ pm,j)

]

Now consider the following class of formulas (where pii,j abbreviates pi,j ∨ · · · ∨ pi,j with i copies of pi,j):

PHP′n =

∨∨

0≤i≤n

∧∧

1≤j≤n

[

pii,j ∨∨∨

i<m≤n

pm,j

]

We have for each n a derivation from PHP′n to PHPn of length O(n3):

PHP′n

= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i≤n

∧∧

1≤j≤n[pii,j ∨

∨∨

i<m≤n pm,j ]n(n+1)/2∗ai↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

∨∨

0≤i≤n

∧∧

1≤j≤n[pii,j ∨

∨∨

i<m≤n([pi,j ∨ pi,j ] ∧ pm,j)]n(n+1)/2∗s −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

∨∨

0≤i≤n

∧∧

1≤j≤n[pii,j ∨ pn−i

i,j∨∨∨

i<m≤n(pi,j ∧ pm,j)]= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

∨∨

0≤i≤n

∧∧

1≤j≤n[pni,j ∨

∨∨

i<m≤n(pi,j ∧ pm,j)]n(n+1)(n−1)∗ac↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

∨∨

0≤i≤n

∧∧

1≤j≤n[pi,j ∨∨∨

i<m≤n(pi,j ∧ pm,j)]n(n+1)∗s −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

∨∨

0≤i≤n[∧∧

1≤j≤n pi,j ∨∨∨

1≤j≤n

∨∨

i<m≤n(pi,j ∧ pm,j)]= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

PHPn

Remark 7.2. Since PHP′n is just an instance of QHQn with qi,j,k = pi,j , every polynomial-size proof of

QHQn yields also a polynomial-size proof of PHPn. On the other hand, with the substitution (found by ananonymous referee)

pi,j 7→∧∧

1≤k≤i

qi,j,k ∧∧∧

i<k≤n

qk,j,i+1

a polynomial-size proof of PHPn can be transformed into a polynomial-size proof of QHQn. Thus the resultby Buss [21] can be used to give a polynomial-size proof of QHQn in SKS.

For a given number n, we define for all 0 ≤ i ≤ n and 1 ≤ j ≤ n the formula

Qi,j =∨∨

1≤k≤i

qi,j,k ∨∨∨

i<k≤n

qk,j,i+1

= qi,j,1 ∨ qi,j,2 ∨ · · · ∨ qi,j,i ∨ qi+1,j,i+1 ∨ qi+2,j,i+1 ∨ · · · ∨ qn,j,i+1

(22)

Then QHQn = (Q0,1 ∧ · · · ∧ Q0,n) ∨ (Q1,1 ∧ · · · ∧ Q1,n) ∨ · · · ∨ (Qn,1 ∧ · · · ∧ Qn,n). The formula Qi,j consistsof n disjuncts. Let Q∨m

i,j denote the formula obtained from Qi,j by removing the mth disjunct. Then forall m ≤ i we have Qi,j = Q∨m

i,j ∨ qi,j,m and for all m > i we have Qi,j = Q∨mi,j ∨ qm,j,i+1. Figure 4 shows a

derivation in sKS from QHQn−1 to QHQn of length O(n3). In that figure, the number z1 is n·(n−1)·(n−2)/2,and z2 is n · (n− 1) · (n− 1). The used substitution is defined as follows:

qi,j,k 7→ [qi,j,k ∨ qn,j,k] ∧ [qn,j,i+1 ∨ qi,n,k] .

Since the proof of QHQ1 is trivial, we exhibited a cut-free polynomial-size proof of QHQn and PHPn. Wecan transform the complete proof of QHQn into an eKS proof by renaming the variables qi,j,k at each stage

12

.

QHQn−1= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i<n

∧∧1≤j<n

[∨∨

1≤k≤iqi,j,k ∨

∨∨i<k<n

qk,j,i+1]sub↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i<n

∧∧1≤j<n

[∨∨

1≤k≤i[(qi,j,k ∧ qn,j,k) ∨ (qn,j,i+1 ∧ qi,n,k)] ∨

∨∨i<k<n

([qk,j,i+1 ∨ qn,j,i+1] ∧ [qn,j,k+1 ∨ qk,n,i+1])]z1∗m −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i<n

∧∧1≤j<n

[∨∨

1≤k≤i([qi,j,k ∨ qn,j,i+1] ∧ [qn,j,k ∨ qi,n,k]) ∨

∨∨i<k<n

([qk,j,i+1 ∨ qn,j,i+1] ∧ [qn,j,k+1 ∨ qk,n,i+1])]z2∗m −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i<n

∧∧1≤j<n

([∨∨

1≤k≤i[qi,j,k ∨ qn,j,i+1] ∨

∨∨i<k<n

[qk,j,i+1 ∨ qn,j,i+1]]∧

[∨∨

1≤k≤i[qn,j,k ∨ qi,n,k] ∨

∨∨i<k<n

[qn,j,k+1 ∨ qk,n,i+1]]= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i<n

∧∧1≤j<n

([qi,j,1 ∨ · · · ∨ qi,j,i ∨ qi+1,j,i+1 ∨ · · · ∨ qn−1,j,i+1 ∨ q n−1n,j,i+1]∧

[qn,j,1 ∨ · · · ∨ qn,j,i ∨ qn,j,i+2 ∨ · · · ∨ qn,j,n ∨ qi,n,1 ∨ · · · ∨ qi,n,i ∨ qi+1,n,i+1 ∨ · · · ∨ qn−1,n,i+1]= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i<n

∧∧1≤j<n

([Q∨ni,j ∨ q n−1

n,j,i+1] ∧ [Q∨i+1n,j

∨ Q∨ni,n])

z2∗ac↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨0≤i<n

∧∧1≤j<n

([Q∨ni,j ∨ qn,j,i+1] ∧ [Q∨i+1

n,j∨ Q∨n

i,n])= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i<n

∧∧1≤j<n

(Qi,j ∧ [Q∨i+1n,j

∨ Q∨ni,n])

= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨0≤i<n

(Qi,1 ∧ · · · ∧ Qi,n−1 ∧ [Q∨i+1n,1

∨ Q∨ni,n] ∧ · · · ∧ [Q∨i+1

n,n−1∨ Q∨n

i,n])n(n−2)∗d↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i<n(Qi,1 ∧ · · · ∧ Qi,n−1 ∧ [(Q∨i+1

n,1∧ · · · ∧ Q∨i+1

n,n−1) ∨ Q∨ni,n])

n∗ai↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨0≤i<n

(Qi,1 ∧ · · · ∧ Qi,n−1 ∧ [(Q∨i+1n,1

∧ · · · ∧ Q∨i+1n,n−1

∧ [qn,n,i+1 ∨ qn,n,i+1]) ∨ Q∨ni,n])

n∗s −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨0≤i<n

(Qi,1 ∧ · · · ∧ Qi,n−1 ∧ [(Q∨i+1n,1

∧ · · · ∧ Q∨i+1n,n−1

∧ qn,n,i+1) ∨ qn,n,i+1 ∨ Q∨ni,n])

= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨0≤i<n

(Qi,1 ∧ · · · ∧ Qi,n−1 ∧ [Qi,n ∨ (Q∨i+1n,1

∧ · · · ∧ Q∨i+1n,n−1

∧ qn,n,i+1)])n∗s −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

[∨∨

0≤i<n(Qi,1 ∧ · · · ∧ Qi,n−1 ∧ Qi,n)] ∨ [

∨∨0≤i<n

(Q∨i+1n,1

∧ · · · ∧ Q∨i+1n,n−1

∧ qn,n,i+1)]= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

[∨∨

0≤i<n

∧∧1≤j≤n

Qi,j ] ∨ [(Q∨1n,1 ∧ · · · ∧ Q∨1

n,n−1 ∧ qn,n,1) ∨ · · · ∨ (Q∨nn,1 ∧ · · · ∧ Q∨n

n,n−1 ∧ qn,n,n)](n−1)(n−1)∗m −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

[∨∨

0≤i<n

∧∧1≤j≤n

Qi,j ] ∨ ([Q∨1n,1 ∨ · · · ∨ Q∨n

n,1] ∧ · · · ∧ [Q∨1n,n−1 ∨ · · · ∨ Q∨n

n,n−1] ∧ [qn,n,1 ∨ · · · ∨ qn,n,n])= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

[∨∨

0≤i<n

∧∧1≤j≤n

Qi,j ] ∨ ([q n−1n,1,1

∨ · · · ∨ q n−1n,1,n] ∧ · · · ∧ [q n−1

n,n−1,1∨ · · · ∨ q n−1

n,n−1,n] ∧ [qn,n,1 ∨ · · · ∨ qn,n,n])z2∗ac↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

[∨∨

0≤i<n

∧∧1≤j≤n

Qi,j ] ∨ ([qn,1,1 ∨ · · · ∨ qn,1,n] ∧ · · · ∧ [qn,n−1,1 ∨ · · · ∨ qn,n−1,n] ∧ [qn,n,1 ∨ · · · ∨ qn,n,n])= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

[∨∨

0≤i<n

∧∧1≤j≤n

Qi,j ] ∨ (Qn,1 ∧ · · · ∧ Qn,n−1 ∧ Qn,n)= −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∨∨

0≤i≤n

∧∧1≤j≤n

Qi,j= −−−−−−−−−−−−−−−−−−−−−−−−−−

QHQn

Figure 4: Derivation from QHQn−1 to QHQn

(see proof of Theorem 6.3) and use the extension formulas5

q′i,j,k ⇔ [qi,j,k ∨ qn,j,k] ∧ [qn,j,i+1 ∨ qi,n,k]

as extension axioms, i.e., the rules

q′i,j,kext↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

[qi,j,k ∨ qn,j,k] ∧ [qn,j,i+1 ∨ qi,n,k]

q′i,j,kext↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

(qi,j,k ∧ qn,j,k) ∨ (qn,j,i+1 ∧ qi,n,k)(23)

In [24], Japaridze provides another cut-free polynomial size proof of PHPn. His system of deep cirquentsuses a form of sharing instead of extension or substitution.

8. Conclusions and future work

This paper provides more new open problems than it provides answers. Figure 5 shows a refined versionof Figure 2 (see also [12]). A solid arrow A ,, B means that A p-simulates B, the notation A × ,, Bmeans that A does not p-simulate B, and a dotted arrow A ,, B means that it is not known whetherA p-simulates B or not. The open problems indicated by these dotted arrows are surprisingly difficult:

5To distinguish between the propositional variable occurrences in QHQn and the occurrences QHQn−1, we use q′ for thosein QHQn−1. This is more legible than adding yet another index to the q.

13

Gentzenwith cuts 11

��

SKS =Frege

��

qq(1) 00

(5)

%%

eSKS = xSKS=Frege + ext.

rr11

��

sSKS =Frege + sub.

pp

��cut-freeGentzen

× ,,

×

SS

KSmm

(2)

SS

(3) ..eKSmm

(4)

KK

(5)

ff

22 sKS

(4)

JJ

(6)rr

Figure 5: Classification of propositional proof systems

(1) The question whether SKS p-simulates eSKS is equivalent to the question whether Frege systems p-simulate extended Frege systems. This question has already been asked in [1], and is one of the mostimportant open problems in the area of proof complexity.

(2) I conjecture that KS does not p-simulate SKS (see also [12] and [25]),

(3) I also conjecture that KS does not p-simulate eKS. More precisely, it is conjectured that KS cannotprovide polynomial size proofs of the formulas PHPn (or QHQn), whereas this is possible in SKS (asshown in [21]) as well as in eKS (as shown in Section 7). However, so far, no technique has beendeveloped for showing that something cannot be done in KS.

(4) This is the question whether extension or substitution can simulate the behavior of the cut. It is oneof the contributions of this paper that this question can now be asked. I conjecture that the answeris positive, but it is not clear how to prove it. Note that the naive cut elimination procedures failin the presence of extension. Even if we manage to modify the technicalities such that we get a cutelimination procedure for eSKS, it is not clear how to avoid the exponential blow-up usually caused bycut elimination.

(5) The questions whether extension without cut is as powerful as the cut without extension, and vice-versa,can be seen as the little brothers of (1).

(6) It has already been shown in [1] that under the presence of cut substitution p-simulates extension, butwithout cut, this question is not trivial.

Remark 8.1. The general cocontraction rule c↑ (see Proposition 3.5) can be reduced to its atomic version

F{a}ac↑ −−−−−−−−−−

F{a ∧ a}

if the medial rule m (see (1)) is present [11]. It has recently been shown [26, 25] that the system KS + ac↑quasi-polynomially simulates SKS, and it is conjectured that this result can be improved to a polynomialsimulation. Furthermore, one can show that sKS (and therefore also eKS) p-simulates KS + ac↑. The tworesults together could provide an answer for one direction of (5). This also raises the question whetherKS+ac↑ can p-simulate eKS. In any case, we have here four ways of proof compression—cocontraction, cut,extension, and substitution—and they can all be studied independently in KS.

Acknowledgments

I thank Paola Bruscoli, Anupam Das, Alessio Guglielmi, and Tom Gundersen for fruitful discussions andhelpful comments improving the readability of the paper.

14

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